Quantum strong energy inequalities

TL;DR

This paper derives quantum energy inequalities for the effective energy density of a non-minimally coupled scalar field, providing a step towards quantum singularity theorems by establishing bounds on negative energy densities.

Contribution

It introduces the first quantum strong energy inequalities for the effective energy density, including state-dependent bounds relevant for quantum singularity theorems.

Findings

Derived difference QEIs for the EED in quantum field theory.

Lower bounds depend on the quantum state and grow slower with temperature.

Bounds are nontrivial and of lower energetic order than the EED.

Abstract

Quantum energy inequalities (QEIs) express restrictions on the extent to which weighted averages of the renormalized energy density can take negative expectation values within a quantum field theory. Here we derive, for the first time, QEIs for the effective energy density (EED) for the quantized non-minimally coupled massive scalar field. The EED is the quantity required to be non-negative in the strong energy condition (SEC), which is used as a hypothesis of the Hawking singularity theorem. Thus establishing a quantum strong energy inequality is a first step towards a singularity theorem for matter described by quantum field theory. More specifically, we derive a difference QEI, where the local average of the EED is normal-ordered relative to the one in a reference state. Furthermore, the lower bounds we derive over timelike geodesics or spacetime volumes turn out to depend on the state of interest. We analyse the state-dependence of these bounds in Minkowski spacetime for thermal (KMS) states, and show that the lower bounds grow more slowly in magnitude than the EED itself as the temperature increases. The lower bounds are therefore of lower energetic order than the EED, and qualify as nontrivial state-dependent QEIs.